Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory. Vassily Olegovich Manturov

First, let us prove that if Δ is a disc diagram over the presentation (1.1) with contour p, its label φ(p) = 1 in the group G.

      If the diagram Δ contains exactly one cell Π, then in the free group F we have either φ(p) = 1 (if Π is a 0-cell) or φ(p) = R^±1 for some R ∈ (if Π is an -cell). In any case, φ(p) = 1 in the group G.

      If Δ has more than one cell, then the diagram can be cut by a path t into two disc diagrams Δ₁, Δ₂ with fewer cells. We can assume that their contours are p₁t and p₂t⁻¹ where p₁p₂ = p. By induction it holds that φ(p₁t) = 1 and φ(p₂t⁻¹) = 1 in the group G. Therefore

      in the group G.

      2) Now let us prove the inverse implication. To achieve it we need for a given word W such that W = 1 in the group G to construct a diagram Δ with contour p such that φ(p) = W.

      It is well-known that in the free group F the word W equals a word for some Ri ∈ .

      Construct a polygonal line t₁ on a plane and mark its segments with letters so that the line reads the word X₁. Connect a circle s₁ to the end of this line and mark it so that it reads if we walk around it clockwise. Now we glue 0-cells to t₁, s₁ and to obtain a set homeorphic to a disc. We obtain a diagram with contour of the form e₁ . . . ek with φ(e₁) ≡ 1 ≡ φ(ek) and .

      Construct the second diagram analogously for the word and glue it to the first diagram by the edge ek.

      Continue the process until we obtain a diagram Δ′ such that φ(∂Δ′) ≡ V, see Fig. 1.3.

      Finally, gluing some 0-cells to the diagram Δ′ we can transform the word V into the word W getting a diagram Δ such that φ(∂Δ) ≡ W. That completes the proof.

      This lemma means that disc diagrams can be used to describe the words in a group which are equal to the neutral element of the group. It turns out that annular diagrams can be used in a similar manner.

      Fig. 1.3The diagram Δ′ with boundary label φ(∂Δ′) ≡ V

Lemma 1.2 (Schupp [Schupp, 1968]). Let V, W be two arbitrary nonempty words in the alphabet ¹. Then they are conjugate in a group G given by its presentation (1.1) if and only if there exists an annular diagram over the presentation (1.1) such that it has two contours p and q with the labels φ(p) ≡ V and φ(q) ≡ W⁻¹.

      Let p be a loop on a surface S such that its edges form a boundary of some subspace homeomorphic to a disc. Then the restriction of the cell partitioning Δ to the subspace is a cell partitioning on the space which is called a submap Γ of the map Δ. Note that by definition a submap is always a disc submap.

      A subdiagram of a given diagram Δ is a submap Γ of the map Δ with edges endowed with the same labels as in the map Δ. Informally speaking, a subdiagram is a disc diagram cut out from a diagram Δ.

      Let us state an additional important result about the group diagrams.

      Lemma 1.3. Let p and q be two (combinatorially) homotopic paths in a given diagram Δ over a presentation (1.1) of a group G. Then φ(p) = φ(q) in the group G.

      In the next section the diagrammatic approach will be used to deal with groups satisfying the small cancellation conditions. In that theory a process of cancelling out pairs of cells of a diagram is useful (in addition to the usual process of cancelling out pairs of letters a and a⁻¹ in a word). The problem is that two cells which are subject to cancellation do not always form a disc submap, so to define the cancellation process correctly we need to prepare the map prior to cancelling a suitable pair of cells. Let us define those notions in detail.

      First, for a given cell partitioning Δ we define its elementary transformations (note that elementary transformations are defined for any cell partitioning, not necessarily diagram).

      Definition 1.5. The following three procedures are called the elementary transformations of a cell partitioning Δ:

      (1)If the degree of a vertex o of Δ equals 2 and this vertex is boundary for two different sides e₁, e₂, delete the vertex o and replace the sides e₁, e₂ by a single side e = e₁ ∪ e₂;

      (2)If the degree of a vertex o of a n–cell Π (n ≥ 3) equals 1 and this vertex is boundary for a side e, delete the side e and the vertex o (the second boundary vertex of the side e persists);

      (3)If two different cells Π₁ and Π₂ have a common side e, delete the side e (leaving its boundary vertices), naturally replacing the cells Π₁ and Π₂ by a new cell Π = Π₁ ∪ Π₂.

      Now we can define a 0-fragmentation of a diagram Δ. First, consider a diagram Δ′ obtained form the diagram Δ via a single elementary transformation. This transformation is called an elementary 0-fragmentation if one of the following holds:

      (1)The elementary transformation is of type 1 and either φ(e₁) ≡ φ(e), φ(e₂) ≡ 1 or φ(e₂) ≡ φ(e), φ(e₁) ≡ 1 and all other labels are left unchanged;

      (2)The elementary transformation is of type 2 and φ(e) ≡ 1;

      (3)The elementary transformation is of type 3 and one of the cells Π₁, Π₂ became a 0-cell.

      Definition 1.6. A diagram Δ′ is a 0-fragmentation of a diagram Δ if it is obtained from the diagram Δ by a sequence of elementary 0-fragmentations.

      Note that 0-fragmentation does not change the number of -cells